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Theorem rlimdm 12025
Description: Two ways to express that a function has a limit. (The expression  (  ~~> r  `  F ) is sometimes useful as a shorthand for "the unique limit of the function  F"). (Contributed by Mario Carneiro, 8-May-2016.)
Hypotheses
Ref Expression
rlimuni.1  |-  ( ph  ->  F : A --> CC )
rlimuni.2  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
Assertion
Ref Expression
rlimdm  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r  `  F ) ) )

Proof of Theorem rlimdm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldmg 4874 . . . 4  |-  ( F  e.  dom  ~~> r  -> 
( F  e.  dom  ~~> r  <->  E. x  F  ~~> r  x ) )
21ibi 232 . . 3  |-  ( F  e.  dom  ~~> r  ->  E. x  F  ~~> r  x )
3 simpr 447 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  x )
4 df-fv 5263 . . . . . . 7  |-  (  ~~> r  `  F )  =  ( iota y F  ~~> r  y )
5 vex 2791 . . . . . . . 8  |-  x  e. 
_V
6 rlimuni.1 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : A --> CC )
76adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F : A --> CC )
8 rlimuni.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
98adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  sup ( A ,  RR* ,  <  )  =  +oo )
10 simprr 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  y )
11 simprl 732 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  x )
127, 9, 10, 11rlimuni 12024 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  y  =  x )
1312expr 598 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  ->  y  =  x ) )
14 breq2 4027 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
153, 14syl5ibrcom 213 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( y  =  x  ->  F  ~~> r  y ) )
1613, 15impbid 183 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  <->  y  =  x ) )
1716adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( F 
~~> r  y  <->  y  =  x ) )
1817iota5 5239 . . . . . . . 8  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( iota y F  ~~> r  y )  =  x )
195, 18mpan2 652 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( iota y F  ~~> r  y )  =  x )
204, 19syl5eq 2327 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
213, 20breqtrrd 4049 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r  `  F ) )
2221ex 423 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F ) ) )
2322exlimdv 1664 . . 3  |-  ( ph  ->  ( E. x  F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F
) ) )
242, 23syl5 28 . 2  |-  ( ph  ->  ( F  e.  dom  ~~> r  ->  F  ~~> r  (  ~~> r  `  F ) ) )
25 rlimrel 11967 . . 3  |-  Rel  ~~> r
2625releldmi 4915 . 2  |-  ( F  ~~> r  (  ~~> r  `  F )  ->  F  e.  dom  ~~> r  )
2724, 26impbid1 194 1  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r  `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   dom cdm 4689   iotacio 5217   -->wf 5251   ` cfv 5255   supcsup 7193   CCcc 8735    +oocpnf 8864   RR*cxr 8866    < clt 8867    ~~> r crli 11959
This theorem is referenced by:  caucvgrlem2  12147  caucvg  12151  dchrisum0lem3  20668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-rlim 11963
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